Molecular modelling of entangled polymer fluids under flow The ...

3.2. SINGLE MODE POM-POM MODEL 55
2
1.8
γ . [sec -1 ]
backbone stretch, λ
1.6
1.4
3
1
0.3
0.1
1.2
1
1 10 100
time [sec]
Figure 3.5: Evolution of backbone stretch for a simple shear flow. τ b = 3sec, τ s = 1sec
and q = 5.
by the size of this term relative to the stretch relaxation of the molecule. In simple
shear S xy tends a constant value of
S xy →
τ b ˙γ
3 + 2 ˙γ 2 τ 2 b
as t → ∞. (3.9)
Thus, in the limit τ b ˙γ ≫ 1, S xy
becomes,
→ 1/2 ˙γτ b for large t and so the stretch equation
d
dt λ = [ 1
2τ b
− 1 τ s
]
λ + 1 τ s
. (3.10)
Since the pom-pom model requires that τ b > τ s , the stretch will always tend to an
equilibrium value of
1
λ = , (3.11)
1 − τs
2τ b
[ ] 1
− 2τ
[Inkson et al. (1999)]. Any additional transient stretch decays as e
1 t b τs towards this
equilibrium value. In the simple shear case the equilibrium value of λ is independent of
˙γ in the ˙γ ≫ 1/τ b limit, hence sustained stretching will never be produced regardless
of the size of the shear rate.
In exponential shear S xy decays as
S xy → 2ατ b + 1
2ατ b
e −αt t → ∞. (3.12)
Note that this behaviour is only weakly dependent on the orientation relaxation time
since at long times the shear rate is large enough for the deformation to be essentially